Copyright and personal use
This question has no single answer, because copyright law differs by country. What follows is the general shape of it, not legal advice.
Two separate questions
The first is what the law in your country says about making a copy for yourself. The second is what the platform’s own terms of service say. They are different things: a platform’s terms are a contract with you, not legislation.
Most discussion online confuses the two, which is why the same question gets opposite answers depending on who is answering.
Where you are clearly fine
Content you created and uploaded yourself. Content published under a Creative Commons licence that permits copying — YouTube marks these in the video description. Content in the public domain. And anything where you have the rights holder’s permission.
These four cases cover a great deal of what people actually want to save, and none of them is ambiguous.
Where it gets uncertain
Some countries have a private-copying or fair-dealing exception that covers a personal copy of material you already have lawful access to. Others do not, or limit it narrowly. The exception rarely extends to sharing, re-uploading or any commercial use.
If a copy matters to you and the answer is not obvious, the safe route is to ask the rights holder. Creators are frequently willing when someone simply asks.